Property REF  is sometimes called “robust transversality,” and it asserts that for each \(p\in P\) , a typical pair of lines incident to \(p\)  point in well-separated directions. With this condition, Proposition REF  can also be viewed as a bilinear incidence theorem. Thus our reduction from Proposition REF  to REF  will employ “bilinear to linear” type arguments, which were first used by Tao, Vargas, and Vega in the context of the Kakeya and Restriction problems [1]}. A prototype for the specific arguments used here can be found in [2]}. See also [3]} for a similar argument in a slightly different setting.
